A commuter travels to work along a road with many traffic signals. After stoppin

Question

A commuter travels to work along a road with many traffic signals. After stopping at one red light the commuter accelerates at 2 m/s2. Just as he passes a speed camera, he notices that the next traffic light has turned red and he must stop, which he does by reversing his acceleration and now decelerating at a constant rate of 2 m/s2; coming to a stop just at the traffic light. The distance between the two traffic lights is 800 m.

A. How does the time he is speeding up compare to the time he is slowing down? Explain your reasoning.

B. How much time does he take to get from the first traffic light to the seconds? Explain your reasoning.

C. The speed limit on the road is 30 mi/h. If he is going more than about 15 mi/h faster than that, hell get a traffic ticket in the mail. Do you expect him to get one? Explain why you think so. Show by explicit calculation. (Useful conversion factor: 1 m/s = 2.24 mi/h.)

Explanation / Answer

A. since the speeding up and speeding down acceleration is the same ( with opposite signs)
   and while speeding up, initial velocity is 0, anf final velocity is v
   while speeding down final velocity is 0, and initial velocity is v
   so, from symmetry, the timne spend speeding up and down is the same
B. time taken to move 800 m be t
   time taken to speed up = t1
   time taken to speed down = t2 = t1
   trime in constant speed v = t3
   then t = t1 + t2 + t3
   so, from equaitons of kinematics
   v = at1 = 2t1
   and 2*a*s1 = v^2

   so 2s1 + vt3 = 800
   v^2/a + vt3 = 800
   2t1^2 + 2t1*t3 = 800
   t1(t1 + t3) = 400
   t3 = (400 - t1^2)/t1

   so t = 2t1 + (400 - t1^2)/t1 = 2t1 + 400/t1 - t1 = t1 + 400/t1
   hence if he accelerated for time t1
   then it takes him a total of t1 + 400/t1 s to cover 800 m
C. lets assume he reaches max speed 45 mi/h = 45/2.24 = 20.089 m/s
   then distance covered to reach this speed, s = v^2/2a = 20.089^2/2*2 = 100.89 m
   hence thi is less than 800/2 = 400 m
   so the person can get a traffic ticket if he accelerates for 100 m

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